Array Research A Research Example THE START Pace

Array Research: A Research Example THE START Pace University 325 Goldstein AC PVL & 416 A WP GC rfrank @ pace. edu 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 1

Table of Contents 1/3 ☺ ☺ ☺ ☺ TOC [2 -4] [5 -7] [8 -10] [11 -13] [14 -16] [17 -19] 3 69 (27 Head, 42 text) Research “Tree” Flow Diagram Observation & Binomial Theorem Scalars & 1 -D Vectors Outer Products & 0 -D Null Arrays Master Equation Mystery Solved ☺ Catenation (Try 1) 3 3 3 [20 -22] 3 [23 -25] 3 24 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 2

Table of Contents 2/3 ☺ ☺ Analysis of Catenation The Law of Catenation Continuous Arrays Negative Length Arrays & Dual View Complex (& Imaginary) Length Fractional Dimensions P(F(A))= F(P(A)) ☺ ☺ ☺ Given P(A), Find A. (Roots) 3 3 3 [25 -27] [28 -30] [31 -33] [34 -37] [38 -40] [41 -43] [44 -47] 4 3 3 4 [48 -50] 3 26 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 3

Table of Contents 3/3 ☺ ☺ ☺ Structure Topology Questions Catenation and the Big Bang Alternative Summary Appendix I: The Binomial Theorem Appendix II: The Binomial Coefficients Appendix III: Some Coefficients of (1+X)1/2 ☺ ☺ Appendix IV: Detail of “Big Bang” 3 [51 -53] [54 -56] 3 [57 -58] [59 -60] [61 -63] [64 -65] 2 2 3 2 [66 -69] 4 19 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 4

The “Tree” Flow of the Research 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 5

Discovered “Array Structure Characteristic Polynomial” Polynomial == “Polynomial Form” 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 6

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The Observation, & Binomial Theorem 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 8

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Basic Representation Idea Basic Binomial Theorem Idea 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 10

All Dimension Scalars Are = & The Canonical 1 -D 1 -Vector Array 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 11

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Derivation of RECURSION Derivation of a 1 -D, N-vector polynomial form RECURSION 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 13

Outer Product , Null Arrays, {2, 3} Example Arrays are Made by Outer Product of 1 -D k-Vectors (The ki are the elements of the shape vector) The Algebra of Null Arrays & A {2, 3} Array Structure Example 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 14

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CREATING A {2, 3} ARRAY BY THE OUTER PRODUCT OF VECTORS Polynomial Multiplication This means 6 Cells (0 -D Scalars), 5 1 -D Vectors, 1 (2 -D) Thingie. 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 16

The Master Equation 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 17

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Sum over all combinations of N-j factors. {2, 3, 4} Example 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 19

Mystery Solved 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 20

{ }, {1, 1}, {1, 1, 1} Contents Counts Are The Same =1 All Structure Polynomials Start With The structure of an array is independent of the mapping to the co-domain that determines the cell contents. 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 21

First Try at Catenation 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 22

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Analysis of a Catenation Example 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 25

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The anti glue must be subtracted from catenations And added (glued) into decatenations. 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 27

The Law of Catenation 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 28

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The anti glue must be subtracted from catenations And added (glued) into decatenations. 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 30

Continuous Arrays 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 31

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Negative Length Arrays & the Dual View Negative Length Arrays 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 34

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Complex (or Imaginary) Length Arrays 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 38

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Fractional Dimension Arrays (The Square Root of a 1 -Vector) [New results on shape vectors adds to this] 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 41

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Infinite dimensionality 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 43

P(F(A)) = F(P(A)) Array Structure of Functions of Arrays [New results on shape vectors adds to this] 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 44

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Given P(A) Find A. Observation on “Defective” Arrays 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 48

Observation: "Defective" arrays in ordinary space often have polynomials that lie in more complicated spaces. 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 49

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Questions About the Meaning of Array Representations – The Topology of Array Structure 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 51

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Same Array!! 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 53

Catenation and the Big Bang (Alternative) Catenation applied to Steinhardt & Turok idea. 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 54

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Appendix: The Binomial Coefficients SUMMARY 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 57

Summary ☺ Example of following two ideas ☺ Discover new things ☺ ☺ ☺ Uncover unexpected connections Get new ideas on related connections Don’t limit your view to a “discipline” Keep notebooks Write something every day ☺ Use your unconscious ☺ Keep at it in spite of the “experts” (De. Broglie) 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 58

Appendix: The Binomial Coefficients Appendix I The Binomial Theorem 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 59

Binomial Theorem 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 60

Appendix: The Binomial Coefficients Appendix II The Binomial Coefficients 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 61

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Appendix: The Binomial Coefficients Appendix III Some coefficients of (1+X)1/2 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 64

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Appendix: The Binomial Coefficients Appendix IV Details of “Big Bang” 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 66

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Array Research: A Research Example THE END 2/13/2022 Array Research: A Research Example V 11 (C) Ronald I. Frank 2008 70